The curvature tag has no usage guidance.

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### The Laplacian of an expression involving the Ricci tensor

While doing some computations on a compact Riemannian manifold I have reached the following expression:
$$ \Delta_y \big( Ric_y (\exp_y ^{-1} x, \exp_y ^{-1} x) \big) (x)$$
where $\Delta_y$ is the ...

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### Negative curvature in the middle of $R^{3}$

What's a simple example of metric on $R^{3}$ which has negative scalar curvature inside of a (limited) set N, and is equal to the standard (Euclidean) metric outside?
Basically, I am asking for a ...

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### Strake's splitting theorem for complex sectional curvature

Strake's splitting theorem: Let $(M^n,g)$ be an open manifold with sectional curvature $K\ge 0$ and soul $\Sigma^k$. If the normal holonomy group of $\Sigma$ is trivial, $M$ splits isometrically into ...

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### Bounding distance between geodesics in manifolds with nonpositive curvature

This is a duplicate of a question at the stackexchange which was not answered. I've recently read (in some notes by Mark Pollicott) the following related claims, which, although quite intuitive, I ...

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### Normal fields of geodesic spheres

This question is related to this one (http://math.stackexchange.com/questions/1383511/normal-curvature-of-geodesic-spheres) I've asked at math.stackexchange. Let $(M,g)$ be a compact Riemannian ...

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### Curvature tensor for a singular manifold

Given a manifold $M$ with its tangent space $TM$ and frame vector field $e \in TM$. However, the transition functions in this tangent bundle are non-smooth. Therefore, the Lie derivative of $e$ with ...

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### Spherical cap is the only compact constant mean curvature surface bounded by a circle

I would like to see that the only compact rotationally invariant constant mean curvature surfaces with boundary a planar circle, are either a planar disk or a spherical cap.
This is stated in the ...

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### Exposition of the Calabi complex

I am interested in a complex derived by Eugenio Calabi in his article "On compact Riemannian manifolds with constant curvature".
The complex is referenced as "Calabi complex" in various citing ...

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### Formula for the curvature of an induced connection

Let $\pi_P:P\to M$ be a principal $G$-bundle on a manifold $M$ endowed with a connection $A$ and $\pi_Q:Q\to M$ be a principal $H$-bundle on a manifold $M$. Let $f:P\to Q$ be a morphism of bundles ...

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### What is the relation between two Riemannian metrics with the same Riemannian curvatures?

Consider two flat Riemannian metrics on a manifold. What is the general relation between these two metrics if the manifold is not simply-connected? What is the answer if two Riemannian metrics have ...

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### Space with $Ric \geq -(n-1)$

Note that hyperbolic space $H$ has $Ric=-(n-1)$. I want to know :
Question : Does there exists a simply connected open complete Riemannian manifold $M$ s.t.
(1) $ Ric\geq -(n-1)$ on $M$
(2) $ ...

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### Integral of Square of Mean Curvature

Let us assume $\text{H}$ is the mean curvature of a compact surface in $\mathbb E^3$ and $g$ is its genus.
When $g$ is arbitrary, we have $\int_{\mathbb S^2}\text{H}^2dV=4\pi$ and ...

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### General form of a metric affine connection with zero curvature

I have read somewhere that the general form of a metric affine connection whose Riemann curvature is zero is given by
$$\Gamma^{i}_{ij}=A^{i}_{\alpha}\partial_{j}A^{\alpha}_{k}$$
where ...

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### Curvature in geometry-interpretation

Previously this question was asked on stack exchange: the answer contained only reference to the wikipedia page which I already read (as mentioned in my post). So here is the question:
The are ...